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Sec. 4.1: Antiderivatives and Indefinite Integration

Sec. 4.1: Antiderivatives and Indefinite Integration. Suppose. What is . What is . What is . Suppose . Suppose . Suppose . Sec. 4.1: Antiderivatives and Indefinite Integration. In other words, what is . What is . Sec. 4.1: Antiderivatives and Indefinite Integration.

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Sec. 4.1: Antiderivatives and Indefinite Integration

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  1. Sec. 4.1: Antiderivatives and Indefinite Integration

  2. Suppose What is What is What is Suppose Suppose Suppose Sec. 4.1: Antiderivatives and Indefinite Integration In other words,what is What is

  3. Sec. 4.1: Antiderivatives and Indefinite Integration Notice that each original differs only in the constant. If we started with we wouldn’t know which constant to use. To take care of this, the constant C is used. If then

  4. Sec. 4.1: Antiderivatives and Indefinite Integration Finding an antiderivative (Toonism: “Antidifferentiatin’”) answers the question, WHO’S YOUR DADDY?

  5. Ex: Ex: Ex: Ex: Sec. 4.1: Antiderivatives and Indefinite Integration Who’s your daddy? Who’s your daddy? Who’s your daddy? Who’s your daddy?

  6. Sec. 4.1: Antiderivatives and Indefinite Integration

  7. Sec. 4.1: Antiderivatives and Indefinite Integration When you solve a differential equation of the form it‘s convenient to write it in the equivalent form The operation of finding all solutions is called indefinite integration.

  8. Sec. 4.1: Antiderivatives and Indefinite Integration x is the variable of integration Notation C is the constant of integration f (x) is called the integrand

  9. Sec. 4.1: Antiderivatives and Indefinite Integration Ex:

  10. Sec. 4.1: Antiderivatives and Indefinite Integration Ex:

  11. AP Calculus BCFriday, 15 November 2013 • OBJECTIVETSW use integration by substitution. • Tests are not graded. • DUAL CREDIT REGISTRATION • Mr. Anderson needs your purple registration form TODAY!!! • ASSIGNMENT DUE MONDAY • Sec. 4.1

  12. Sec. 4.1: Antiderivatives and Indefinite Integration Ex:

  13. Sec. 4.1: Antiderivatives and Indefinite Integration Ex:

  14. Sec. 4.1: Antiderivatives and Indefinite Integration Ex:

  15. Sec. 4.1: Antiderivatives and Indefinite Integration Ex: Find such that and

  16. Sec. 4.1: Antiderivatives and Indefinite Integration Ex: Find the particular solution:

  17. Sec. 4.1: Antiderivatives and Indefinite Integration Ex: Find the particular solution:

  18. Sec. 4.1: Antiderivatives and Indefinite Integration Ex: Find the particular solution:

  19. Sec. 4.1: Antiderivatives and Indefinite Integration The graph of f ' is shown. Graph f. f'

  20. Sec. 4.1: Antiderivatives and Indefinite Integration The graph of f ' is shown. Graph f. f'

  21. Sec. 4.1: Antiderivatives and Indefinite Integration The graph of f " is shown. Graph f. f "

  22. Sec. 4.1: Antiderivatives and Indefinite Integration Ex: The rate of growth dP/dt of a population of bacteria is proportional to the square root of t, where P is the population size and t is the time in days (0 ≤ t ≤ 10). That is, The initial size of the population is 500. After 1 day the population has grown to 600. Estimate the population after 7 days.

  23. Sec. 4.1: Antiderivatives and Indefinite Integration “The initial size of the population is 500.” When t = 0, P = 500.

  24. Sec. 4.1: Antiderivatives and Indefinite Integration “After 1 day, the population has grown to 600.” When t = 1, P = 600.

  25. Sec. 4.1: Antiderivatives and Indefinite Integration “Estimate the population after 7 days.” t = 7 ~2352 bacteria

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